So this is what I’ve been work­ing on this week. I’ve been look­ing at the ori­ent­a­tions of Greek temples. There is an idea that Greek temples always face east, and that’s what I’m test­ing at the moment. If I can show that Greek temples do face East then things get inter­est­ing. This is because in Sicily in the first mil­len­nium BC the nat­ives take on a lot of Greek mater­ial. If I can show that the nat­ives are still prac­tising their own reli­gions in their own way, then I have strong argu­ment that they’re using Greek pot­tery and so on for their own pur­poses rather than simply becom­ing Greeks them­selves. I have res­ults and I’m try­ing to put them together meaningfully.

A lot of the sig­ni­fic­ance depends on the data set and how I use it. For instance if I have only one temple and it points East, that doesn’t really mean a lot. It has to point some­where, so why should that be spe­cial? It could face that way by chance. If I have two temples facing East then that’s a bit bet­ter, but it’s still hardly impress­ive. At it hap­pens I have meas­ure­ments for forty-two temples, but not all of them face East. So are my res­ults sig­ni­fic­ant? Below is me try­ing to work this out and come up with a bet­ter answer than: “Yes, because I say so.” It fol­lows quite a few other chapters in the thesis, so it might not all make sense, but it should make enough sense for people to point out any obvi­ous mis­takes in my hand­ling of probability.

Some of the for­mu­lae may seem a little odd, but hope­fully they’re clear enough. I’ll have to get to grips with MathML to gen­er­ate some for­mu­lae graph­ics for the actual print. This is very much first draft mater­ial.

Given the ori­ent­a­tion of any indi­vidual temple, it is impossible to say pre­cisely purely from stat­ist­ics what the motive for its ori­ent­a­tion was. By its very nature any temple must have an open­ing facing in one dir­ec­tion and so who is to say that it was aligned inten­tion­ally rather than by chance? How­ever, it is pos­sible to quantify the effects of chance when the data are aggreg­ated. For the first example I shall tackle the ques­tion “Do Greek Temples Face East?” For the sake of this ques­tion I am assum­ing any temple which faces between 0° through 90° to 180° is facing East and that any temple facing between 180° through 270° to 0° is facing west. I should also cla­rify that 0° is due North as Pen­rose took his 0° ref­er­ence to be due South.

In the absence of any a pri­ori assump­tions, the chance of any indi­vidual temple facing East should be 0.5, where 0 is com­plete impossib­il­ity and 1 is abso­lute cer­tainty. This is a well known prob­lem in prob­ab­il­ity akin to coin toss­ing and so it is easy to cal­cu­late what, on aver­age will be the res­ult if forty-two temples are aligned purely by chance. The rel­ev­ant for­mula is xbar = 0.5n. In this case n is forty-two and so on aver­age twenty-one temples will face East. In these res­ults forty-one temples face the east­ern hori­zon, and so this would appear to be sig­ni­fic­ant. The ques­tion usu­ally asked is how sig­ni­fic­ant is this?

The com­bin­a­tions of pos­sible res­ult expand rap­idly as more temples are added to the test set. For example if we have just one temple in our test set then there are just two pos­sible outcomes.

How­ever if we add a second temple four out­comes are possible.

If we add a fur­ther temple to make a set of three then there are eight poten­tial outcomes

To com­plete the example a fourth temple will provide six­teen poten­tial combinations:

So the num­ber of com­bin­a­tions of poten­tial East-West ori­ent­a­tions in a set of n temples is 2ⁿ. For the com­plete set of forty-two temples are 2⁴² poten­tial com­bin­a­tions by chance. This is a large fig­ure: four tril­lion three hun­dred ninety-eight bil­lion forty-six mil­lion five hun­dred eleven thou­sand one hun­dred and four com­bin­a­tions are pos­sible. That all bar one of these will face East as in the res­ults of the sur­vey will only occur in forty-two of these com­bin­a­tion as the aber­rant temple could be the first, second, third…forty-second temple meas­ured. Simplist­ic­ally I could argue that the odds that these res­ults were due to chance were 42/4,398,046,511,104 which reduces to 21/2,199,023,255,552. This is approx­im­ately one in a hun­dred mil­lion. This is often the way such odds are cal­cu­lated in archae­ology (note to self add sev­eral ref­er­ences). How­ever there is a super­fi­cial flaw.

If all the temples had faced East then this also would have been con­sidered sig­ni­fic­ant. There­fore I should not be cal­cu­lat­ing the odds that forty-one temples faced East, but rather that at least forty-one temples faced East. This adds one fur­ther com­bin­a­tion, namely EEEEEE…EEEEEE to the set. Per­haps it would be bet­ter to quote odds of 43 to 4,398,046,511,104 which is still approx­im­ately one in a hun­dred mil­lion. Yet this ques­tion can also be asked if fewer temples matched the tested align­ment. What would this chapter have looked like if only forty temples had faced East?

There are con­sid­er­ably more com­bin­a­tions of where two temples in the set face West. The aber­rant temples could be the first and second temples meas­ured, or the first and third or the first and fourth, through to the forty-first and forty-second. There are in fact eight hun­dred and sixty-one com­bin­a­tions. This means that forty temples would be sig­ni­fic­ant then the odds of sim­ilar res­ults being due to chance are 904 to 4,398,046,511,104. This would still be an impress­ively improb­able fig­ure, but this line of reas­on­ing raises the ques­tion of whether thirty-nine temples would be sig­ni­fic­ant or thirty-eight. This is a con­sid­er­ably more ser­i­ous flaw. Com­plic­at­ing the mat­ter fur­ther had forty-one of forty-two temples poin­ted West it would be equally unlikely. Rather than decide what is sig­ni­fic­ant a pos­teri­ori it would be more help­ful if it were pos­sible to decide a pri­ori what would be con­sidered significant.

The usual fig­ure adop­ted in psy­cho­lo­gical tests, where res­ults are prone to be products of chance, is to assume that any­thing below a 5% prob­ab­il­ity is sig­ni­fic­ant. This is par­tic­u­larly suit­able for applic­a­tion here as 5% is an eas­ily cal­cul­able fig­ure. Ninety-five per cent of res­ults in a stand­ard bell curve will be within two stand­ard devi­ations of the mean. Cal­cu­la­tion of the stand­ard devi­ation of a bino­mial prob­ab­il­ity is eas­ily cal­cu­lated via a formula.

where n is the num­ber of temples in the set, p is the prob­ab­il­ity of the temple facing the tar­get range and q is the prob­ab­il­ity of the temple fall­ing out­side this range. This would be equal to 1-p.

For this example, if I were to build thou­sand of sets of forty-two temples, align­ing them ran­domly, the aver­age num­ber of temples in each set facing East would be 21 and 95% of them would have 21+/-6.48074 temples. There­fore for the pur­poses of this study any res­ult between four­teen and twenty-eight temples would have been con­sidered significant.

This is unques­tion­ably con­sid­er­ably less emo­tion­ally sat­is­fy­ing than announ­cing the prob­ab­il­ity of the res­ults being due to chance are a hun­dred mil­lion to one but, given the small data set, it is con­sid­er­ably more defens­ible. The prob­lem is that it is not suf­fi­cient that the res­ults sat­isfy this one in twenty rule. Richard Feyn­man has argued that this rule would logic­ally mean that one in twenty psy­cho­lo­gical laws were stat­ist­ical flukes. This is not entirely fair, such res­ults would sub­ject to fur­ther tests in Psy­cho­logy. How­ever I simply apply this cri­terion to a series of tests then, if I have more than twenty tests, by sheer chance I should expect one ‘sig­ni­fic­ant’ res­ult. There is in par­tic­u­lar the tend­ency to pur­sue tests until some­thing sig­ni­fic­ant is found. Feynman’s cri­ti­cism would then apply to this work. There­fore I pro­pose that this 95% cri­terion is only used as a fil­ter. No mat­ter how seduct­ive a pro­posal is, if it does not meet this cri­terion I should con­clude that the data is insuf­fi­cient to sup­port the hypo­thesis. If the pro­posal sur­passes this cri­terion then there is still an imper­at­ive to provide a his­tor­ical jus­ti­fic­a­tion for the res­ult. In this case the res­ult, only one temple faces away from East appears com­pel­ling, but that means there is a need to explain why that temple – the Hekataion at Selin­unte – is facing con­trary to the other temples.

The Hekataion at Selin­unte is ded­ic­ated to Hekate Tri­formis. Tri­formis refers to the three aspects of the god­dess. She was Artemis, Selene and Hekate. In the form of Hekate she was the dark moon around the cres­cent. She was known as the opener of the door to Hades (check Cash­ford) and so played a role in the quest to recover Persephone from Hades. This is sig­ni­fic­ant as the Hekataion is part of a com­plex at Selin­unte, the other two major gods there being Demeter Mal­aphoros and Zeus Melikhios who both played a role in the mys­ter­ies of Eleusis. By facing the set­ting side of the sky, this ori­ent­a­tion could be inten­tion­ally con­trary to the typ­ical cult prac­tices. How­ever it should also be noted that des­pite being a lunar god­dess the temple points too far to the north to face the most north­erly moon­sets. This would there­fore not be any form of celes­tial align­ment but a con­scious rejec­tion of celes­tial align­ment. This explan­a­tion is plaus­ible without invok­ing new prop­er­ties for Hekate and would be con­sist­ent with the local geo­graphy and con­text of the site.

It would seem jus­ti­fi­able to there­fore claim that Greek temples typ­ical face east.

The next part will fol­low tomorrow.